In this set of notes I want to sketch Gödel’s proof that is consistent with the other axioms of set theory. Gödel’s argument goes well beyond this result; his identification of the class of constructible sets eventually led to the development of inner model theory, one of the main areas of active research within set theory nowadays.

A good additional reference for the material in these notes is Constructibility by Keith Devlin.

1. Definability

The idea behind the constructible universe is to only allow those sets that one must necessarily include. In effect, we are trying to find the smallest possible transitive class model of set theory.

is defined as

where for limit, and where

The first question that comes to mind is whether this definition even makes sense. In order to formalize this, we need to begin by coding a bit of logic inside set theory. The recursive constructions that we did at the beginning of the term now prove useful.

Let me begin with a couple of comments that may help clarify some of the results from last lecture.

First, I want to show a different proof of Lemma 21.2, that I think is cleaner than the argument I gave before. (The argument from last lecture, however, will be useful below, in the proof of Kunen’s theorem.)

Lemma 1 If is measurable, is a -complete nonprincipal ultrafilter over and is the corresponding ultrapower embedding, then

Proof: Recall that if is Mostowski’s collapsing function and denotes classes in then To ease notation, write for

Let Pick such that for all

Lemma 2 With notation as above, for any

Proof: For a set let denote the function constantly equal to Since is an isomorphism, ‘s lemma gives us that the required equality holds iff

but this last set is just

From the nice representation just showed, we conclude that for all But for any such because by Lemma 21 from last lecture. Hence, which is obviously in being definable from and

The following was shown in the proof of Lemma 20, but it deserves to be isolated.

Lemma 3 If is a normal nonprincipal -complete ultrafilter over the measurable cardinal then i.e., we get back when we compute the normal measure derived from the embedding induced by

Finally, the construction in Lemma 10 and preceeding remarks is a particular case of a much more general result.

Definition 4 Given and an ultrafilter over the projection of over is the set of such that

Clearly, is an ultrafilter over

Notice that if is a partition of into sets not in and is given by the unique such that then is a -complete nonprincipal ultrafilter over (Of course, is possible.)

For a different example, let be a -complete nonprincipal ultrafilter over the measurable cardinal and let represent the identity in the ultrapower by Then is the normal ultrafilter over derived from the embedding induced by

Definition 5 Given ultrafilters and (not necessarily over the same set), say that is Rudin-Keisler below in symbols, iff there are sets and a function such that

Theorem 6 Let be an ultrafilter over a set and an ultrafilter over a set Suppose that Then there is an elementary embedding such that

Proof: Fix and for which there is a map such that Clearly, as witnessed by the map and similarly so it suffices to assume that and

Given let be given by Then is well-defined, elementary, and

In effect, iff iff iff where the second equivalence holds by assumption, and it follows that is well-defined.

If denotes the function with domain and constantly equal to then for any since by definition of the map This shows that

Elementarity is a straightforward modification of the proof of Lemma 10 from last lecture.

One can show that Theorem 6 “very nearly” characterizes the Rudin-Keisler ordering, see for example Proposition 0.3.2 in Jussi Ketonen, Strong compactness and other cardinal sins, Annals of Mathematical Logic 5 (1972), 47–76.

Let me begin with a remark related to the question of whether . We showed that this is the case if for some , or if is Dedekind-finite.

Theorem.The axiom of choice is equivalent to the statement that any Dedekind-infinite cardinal is a square.

Proof. Let be a set. Assuming that every D-infinite cardinal is a square, we need to show that is well-orderable. We may assume that . Otherwise, replace with . Let . Assume that is a square, say . Then . By Homework problem 2, , so for some , and .

Lemma.Suppose are D-infinite sets and is an (infinite) initial ordinal. If then either or .

Proof. Let be an injection. If there is some such that we are done, so we may assume that for all there is some such that . Letting be the least such , the map is an injection of into .

By the lemma, it must be that either or else . The former is impossible since , so is well-orderable, and thus so is , and since , then is well-orderable as well.

The only reference I know for precisely these matters is the handbook chapter MR2768702. Koellner, Peter; Woodin, W. Hugh. Large cardinals from determinacy. In Handbook of set theory. Vols. 1, 2, 3, 1951–2119, Springer, Dordrecht, 2010. (Particularly, section 7.) For closely related topics, see also the work of Yong Cheng (and of Cheng and Schindler) on Harr […]

As other answers point out, yes, one needs choice. The popular/natural examples of models of ZF+DC where all sets of reals are measurable are models of determinacy, and Solovay's model. They are related in deep ways, actually, through large cardinals. (Under enough large cardinals, $L({\mathbb R})$ of $V$ is a model of determinacy and (something stronge […]

Throughout the question, we only consider primes of the form $3k+1$. A reference for cubic reciprocity is Ireland & Rosen's A Classical Introduction to Modern Number Theory. How can I count the relative density of those $p$ (of the form $3k+1$) such that the equation $2=3x^3$ has no solutions modulo $p$? Really, even pointers on how to say anything […]

(1) Patrick Dehornoy gave a nice talk at the Séminaire Bourbaki explaining Hugh Woodin's approach. It omits many technical details, so you may want to look at it before looking again at the Notices papers. I think looking at those slides and then at the Notices articles gives a reasonable picture of what the approach is and what kind of problems remain […]

It is not possible to provide an explicit expression for a non-linear solution. The reason is that (it is a folklore result that) an additive $f:{\mathbb R}\to{\mathbb R}$ is linear iff it is measurable. (This result can be found in a variety of places, it is a standard exercise in measure theory books. As of this writing, there is a short proof here (Intern […]

The usual definition of a series of nonnegative terms is as the supremum of the sums over finite subsets of the index set, $$\sum_{i\in I} x_i=\sup\biggl\{\sum_{j\in J}x_j:J\subseteq I\mbox{ is finite}\biggr\}.$$ (Note this definition does not quite work in general for series of positive and negative terms.) The point then is that is $a< x

The result was proved by Kenneth J. Falconer. The reference is MR0629593 (82m:05031). Falconer, K. J. The realization of distances in measurable subsets covering $R^n$. J. Combin. Theory Ser. A 31 (1981), no. 2, 184–189. The argument is relatively simple, you need a decent understanding of the Lebesgue density theorem, and some basic properties of Lebesgue m […]

Given a class $S$, to say that it can be proper means that it is consistent (with the axioms under consideration) that $S$ is a proper class, that is, there is a model $M$ of these axioms such that the interpretation $S^M$ of $S$ in $M$ is a proper class in the sense of $M$. It does not mean that $S$ is always a proper class. In fact, it could also be consis […]

As the other answers point out, the question is imprecise because of its use of the undefined notion of "the standard model" of set theory. Indeed, if I were to encounter this phrase, I would think of two possible interpretations: The author actually meant "the minimal standard model of set theory", that is, $L_\Omega$ where $\Omega$ is e […]